Emmy-Noether-Seminar: Homogeneous varieties with non-reduced stabilizers

Homogeneous varieties with non-reduced stabilizers/b>

Matilde Maccan (Bochum)

Abstract : In this talk, we start with the fundamental question of classifying homogeneous algebraic varieties, over an algebraically closed field. First, we look at the projective case (in the literature, the case of flag varieties), namely quotients of the form G/P, where G is semi-simple and P contains a maximal solvable subgroup. Over the complex numbers, their structure is well-known, while in positive characteristics new objects emerge, due to the Frobenius homomorphism and the existence of non-reduced group (schemes). The classification in characteristic at least five was completed by Wenzel, Haboush and Lauritzen in the 90’s; my contribution consisted in completing the case of small characteristics. Next, focusing on their geometry, we will present a result describing the automorphism group of G/P (as a group scheme), generalizing work of Demazure. Finally, we study a slightly more general class of varieties: the horospherical homogeneous spaces, namely quotients G/H where H contains a maximal unipotent. Quite surprisingly, we find that their classification, in characteristic at least three, is the same as the one in characteristic zero. This last result is joint work with R. Terpereau.